Step 1 :
Let us take A and B as the two positions of the two flag posts and P be the position of the man.
$ \therefore PA + PB = 10$
Hence it is clear that the path described by the man is an ellipse.
The points A and B are the foci.
The length of the major axis is 10 m.
Step 2 :
The equation of the ellipse is of the form $\large\frac{x^2}{a^2}$$+\large\frac{y^2}{b^2}$$=1$
Since $2a=10 \Rightarrow a = 5$
Distance between the foci is 2c = 8
$ \therefore c = 4$
We know $c = \sqrt{a^2-b^2}$
or $ c^2 = a^2-b^2$
Substituting the value for c and a we get,
$16 = 25-b^2$
$ \therefore b^2=9 \Rightarrow b = 3$
Hence the equation of the path traced by the man is
$\large\frac{x^2}{25}$$+\large\frac{y^2}{9}$$=1$